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a given particle reversed, the motions of all other particles being derived from that of the first according to the general law of wave propagation. The relations between components and resultants will evidently not be violated; and if X", 0", E", o", e", Y' denote the streams into which JT, 0', E', o', e', Y' are thus changed, it is evident that the polarizations of X", 0", E", o", e", F" are respectively opposite to those of X, 0, E, o, e, Y. But on account of the reversion in direction of motion it is plain that there is reversion as regards acceleration and retardation of phase, so that in passing from the pair 0", E" to the pair o", e" the phase of 0" is accelerated by p relatively to the phase of E".
Hence the stream X''', polarized oppositely to Xy is resolved into two E", 0"', polarized in the same manner respectively as 0, E, which are recompounded after the phase of the one polarized in the same manner as 0 has been retarded by p relatively to the phase of the other, and the result is Y", a stream polarized in a manner opposite to Y, which proves the theorem.
12. This theorem may be applied to the case of light transmitted through a slice of a doubly refracting crystal, and shews that two streams going in oppositely polarized come out oppositely polarized. Also, since nothing in the demonstration depends upon the order in which the decompositions and recom-positions take place, it is immaterial whether X, X" denote a pair of oppositely polarized incident streams, which give rise to the emergent streams Y, Y", or X, X" denote a pair of oppositely polarized emergent streams, which came from the incident streams F, Y'. A particular case of this theorem was assumed in the preceding article, when it was stated that the pencils, polarized in perpendicular planes, which on coming out of a crystal of calcareous spar are respectively stopped by and transmitted through a Nicol's prism, went into the crystal oppositely polarized.
The theorem will evidently be true of a train consisting of any number of crystalline plates, each possessing the property of resolving the incident light into two oppositely polarized streams, which arc propagated within the medium with different velocities. For two oppositely polarized streams incident on the first plate give rise to two emergent streams which fall oppositely polarized on the second plate, and so on. Since the number